Problem: The lengths of the sides of a non-degenerate triangle are $x$, 13 and 37 units. How many integer values of $x$ are possible?
Explanation: By the triangle inequality, \begin{align*}
x + 13 &> 37, \\
x + 37 &> 13, \\
13 + 37 &> x,
\end{align*} which tell us that $x > 24$, $x > -24$, and $x < 50$.  Hence, the possible values of $x$ are $25, 26, \dots, 49$, for a total of $49 - 25 + 1 = \boxed{25}$.